The best I can think of is to just create one website and register a bunch of domains that each point to separate page on the same site, then create a "minisite" with no internal links on each separate page. It sort of seems like a cantankerous way to get the job done, though. Is there any better or more affordable way to do it?

Many hosting companies will allow you to add as many add-on domains as you wish allowing you to host multiple websites on one hosting account. You can set up multiple websites with different domain names pointing to each add-on site or, as previously suggested, you could use subdomains. Using subdomains, you will not have to pay to register a bunch of domain names.